Let \(A\subset A^{\prime }\)  be an unramified extension of discrete valuation rings (shortly DVR), that is the generator \(x\)  of the maximal ideal of \(A\)  generates also the maximal ideal of \(A^{\prime }\) . Suppose that the induced field extensions \(A/(x)\rightarrow A^{\prime }/xA^{\prime }\) , Fr\((A)\rightarrow \)  Fr\((A^{\prime })\)  are
separable, in other words the inclusion \(A\rightarrow A^{\prime }\)  is regular. Néron [1]} (see here Theorem REF ) proved that every sub-\(A\) -algebra \(B\subset A^{\prime }\)  of finite type can be embedded in a regular subring \(D\subset A^{\prime }\) , essentially of finite type over \(A\) . Moreover, \(D\)  could be chosen contained in Fr\((B)\) , that is \(D\)  could be seen as a desingularization of \(B\)  made with respect of the inclusion \(B\rightarrow A^{\prime }\) .
